Controllability of Second-OrderEquations in 𝐿^𝟐(Ω)

Abstract

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space 𝐿²(Ω): 𝑦̈+𝐴𝑦=1_𝜔𝑢(𝑡), 𝑡∈(0,𝜏], 𝑦(0)=𝑦₀, 𝑦(̇0)=𝑦₁, where Ω is a domain in ℝ^𝑁(𝑁≥1), 𝑦₀,𝑦₁∈𝐿²(Ω), 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢 belongs to 𝐿²(0,𝜏;𝐿²(Ω)), and 𝐴∶𝐷(𝐴)⊂𝐿²(Ω)→𝐿²(Ω) is an unbounded linear operator with the following spectral decomposition: 𝐴𝑧=∑^∞_𝑗=1𝜆_𝑗∑^𝛾𝑗_𝑘=1⟨𝑧,𝜙_𝑗,𝑘⟩𝜙_𝑗,𝑘, with the eigenvalues 𝜆_𝑗 given by the following formula: 𝜆_𝑗=𝑗^2𝑚𝜋^2𝑚, 𝑗=1,2,3,… and 𝑚≥1 is a fixed integer number, multiplicity 𝛾_𝑗 is equal to the dimension of the corresponding eigenspace, and {𝜙_𝑗,𝑘} is a complete orthonormal set of eigenvectors (eigenfunctions) of 𝐴. Specifically, we prove the following statement: if for an open nonempty set 𝜔⊂Ω the restrictions 𝜙^𝜔_𝑗,𝑘=𝜙_𝑗,𝑘|_𝜔 of 𝜙_𝑗,𝑘 to 𝜔 are linearly independent functions on 𝜔, then for all 𝜏≥2/𝜋^𝑚−1 the system is approximately controllable on [0,𝜏]. As an application, we prove the controllability of the 1D wave equation.

1. Introduction

This paper has been motivated by the work in [1] and the articles [2, 3], where a new technique is used to prove the interior approximate controllability of some diffusion process. Particularly in [3], where the authors prove the interior approximate controllability of the following broad class of reaction diffusion equations in the Hilbert space 𝑍=𝐿²(Ω) given by 𝑧′=−𝐴𝑧+1_𝜔𝑢(𝑡),𝑡∈[0,𝜏], (1.1) where Ω is a domain in ℝ^𝑛, 𝜔 is an open nonempty subset of Ω, 1_𝜔 denotes the characteristic function of the set 𝜔, the distributed control 𝑢∈𝐿²(0,𝜏;𝐿²(Ω)) and 𝐴∶𝐷(𝐴)⊂𝑍→𝑍 is an unbounded linear operator with the spectral decomposition𝐴𝑧=^∞_𝑗=1𝜆_𝑗^𝛾𝑗_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (1.2) The eigenvalues 0<𝜆₁<𝜆₂<⋯<⋯𝜆_𝑛→∞ of 𝐴 have finite multiplicity 𝛾_𝑗 equal to the dimension of the corresponding eigenspace, and {𝜙_𝑗,𝑘} is a complete orthonormal set of eigenvectors of 𝐴. The operator −𝐴 generates a strongly continuous semigroup {𝑇_𝐴(𝑡)}_𝑡≥0 given by 𝑇_𝐴(𝑡)𝑧=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡𝛾𝑗}_𝑘=1𝑧,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (1.3) As a consequence of this result, the controllability of the following heat equation follows trivially by putting 𝐴=−Δ𝑧_𝑡=Δ𝑧+1_𝜔𝑢(𝑡,𝑥),in(0,𝜏)×Ω,𝑧=0,on()𝑧()0,𝜏×𝜕Ω,0,𝑥=𝑧₀(𝑥),inΩ. (1.4) Following [1–3], in this paper, we study the interior approximate controllability of the following broad class of second-order equations in the Hilbert space 𝐿²(Ω):𝑦̈+𝐴𝑦=1_𝜔𝑢(𝑡)(],𝑦(0),𝑡∈0,𝜏=𝑦₀,𝑦̇(0)=𝑦₁, (1.5) where the eigenvalues 𝜆_𝑗 of the operator of 𝐴 are given by the following formula:𝜆_𝑗=𝑗^2𝑚𝜋^2𝑚,𝑗=1,2,3,…,𝑚≥1isafixedintegernumber. (1.6) Specifically, we prove the following statement: if for an open nonempty set 𝜔⊂Ω the restrictions 𝜙^𝜔_𝑗,𝑘=𝜙_𝑗,𝑘|_𝜔 of 𝜙_𝑗,𝑘 to 𝜔 are linearly independent functions on 𝜔, then for all 𝜏≥2/𝜋^𝑚−1 the system (1.5) is approximately controllable on [0,𝜏]. Moreover, we can exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time (see Theorem 2.8).

This result implies the interior controllability of the following well-known examples of partial differential equations.

Example 1.1. The 1D Wave Equation 𝑦_𝑡𝑡−Δ𝑦=1_𝜔𝑢(𝑡,𝑥),in(0,𝜏]×Ω,𝑦=0,on()𝑦()0,𝜏×𝜕Ω,0,𝑥=𝑦₀(𝑥),𝑦_𝑡(0,𝑥)=𝑦₁(𝑥),𝑥∈Ω, (1.7) where 𝜔 is an open nonempty subset of Ω=[0,1], 1_𝜔 denotes the characteristic function of the set 𝜔, and the distributed control 𝑢∈𝐿²(0,𝜏;𝐿²(Ω)).

Example 1.2. The Model of Vibrating String Equation 𝑤_𝑡𝑡+Δ²𝑤=1_𝜔𝑢(𝑡,𝑥),in(0,𝜏]×Ω,𝑤=Δ𝑤=0,on()𝑤()0,𝜏×𝜕Ω,0,𝑥=𝜙₀(𝑥),𝑤_𝑡(0,𝑥)=𝜓₀(𝑥),𝑥∈Ω, (1.8) where Ω=[0,1], 𝜔 is an open nonempty subset of Ω, 𝑢∈𝐿²([0,𝜏];𝐿²(Ω)), 𝜙₀,𝜓₀∈𝐿²(Ω).

2. Main Results

In this section, we will prove the main result of this work; to this end, we consider by 𝑋=𝑈=𝐿²(Ω) and the linear unbounded operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 can be written as follows. (a)For all 𝑥∈𝐷(𝐴), we have 𝐴𝑥=^∞_𝑗=1𝜆_𝑗𝐸_𝑗𝜉, (2.1) where 𝐸_𝑗𝑥=^𝛾𝑗_𝑘=1𝜉,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (2.2) So, {𝐸_𝑗} is a family of complete orthogonal projections in 𝑋 and 𝑥=∑^∞_𝑗=1𝐸_𝑗𝑥,𝑥∈𝑋.(b)The semigroup {𝑇_𝐴(𝑡)} generated by −𝐴 can be written as follows:𝑇_𝐴(𝑡)𝑥=^∞_𝑗=1𝑒^{−𝜆𝑗𝑡}E_𝑗𝑥. (2.3)(c)The fractional powered spaces 𝑋^𝑟 are given by 𝑋^𝑟=𝐷𝐴^𝑟=𝑥∈𝑋∶^∞_𝑛=1𝜆_𝑛^2𝑟𝐸_𝑛𝑥²<∞,𝑟≥0, (2.4) with the norm ‖𝑥‖_r=𝐴^𝑟𝑥=^∞_𝑛=1𝜆_𝑛^2𝑟𝐸_𝑛𝑥^21/2,𝑥∈𝑋^𝑟,𝐴^𝑟𝑥=^∞_𝑛=1𝜆^𝑟_𝑛𝐸_𝑛𝑥. (2.5) Also, for 𝑟≥0, we define 𝑍_𝑟=𝑋^𝑟×𝑋, which is a Hilbert space endowed with the norm given by 𝑦𝑣²_𝑍𝑟=𝑦²_𝑟+‖𝑣‖². (2.6)

Proposition 2.1. The operator 𝑃_𝑗∶𝑍_𝑟→𝑍_𝑟, 𝑗≥0, defined by 𝑝_𝑗=𝐸_𝑗00𝐸_𝑗,𝑗≥1, (2.7) is a continuous (bounded) orthogonal projections in the Hilbert space 𝑍_𝑟.

Proof. First we will show that 𝑃_𝑗(𝑍_𝑟)⊂𝑍_𝑟, which is equivalent to show that 𝐸_𝑗(𝑋^𝑟)⊂𝑋^𝑟. In fact, let 𝑥 be in 𝑋^𝑟 and consider 𝐸_𝑗𝑥. Then, ^∞_𝑛=1𝜆_𝑛^2𝑟𝐸_𝑛𝐸_𝑗𝑥²=𝜆_𝑗^2𝑟𝐸_𝑗𝑥²<∞. (2.8) Therefore, 𝐸_𝑗𝑥∈𝑋^𝑟,  for all 𝑥∈𝑋^𝑟.
Now, we will prove that this projection is bounded. In fact, from the continuous inclusion 𝑋^𝑟⊂𝑋, there exists a constant 𝑘>0 such that ‖𝑥‖≤𝑘‖𝑥‖_𝑟,∀𝑥∈𝑋^𝑟. (2.9) Then, for all 𝑥∈𝑋^𝑟, we have the following estimate 𝐸_𝑗𝑥²_𝑟=^∞_𝑛=1𝜆_𝑛^2𝑟𝐸_𝑛𝐸_𝑗𝑥²=𝜆_𝑗^2𝑟𝐸_𝑗𝑥²≤𝜆_𝑗^2𝑟‖𝑥‖²≤𝜆_𝑗^2𝑟𝑘²‖𝑥‖²_𝑟. (2.10) Hence, ‖𝐸_𝑗𝑥‖≤𝜆^𝑟_𝑗𝑘‖𝑥‖_𝑟, which implies the continuity of 𝐸_𝑗∶𝑋^𝑟→𝑋^𝑟. So, 𝑃_𝑗 is a continuous projection on 𝑍_𝑟.

Hence, with the change of variable 𝑦′=𝑣, the system (1.5) can be written as a first-order system of ordinary differential equations in the Hilbert space 𝑍_1/2=𝑋^1/2×𝑋 as follows: 𝑧^′=𝒜𝑧+𝐵_𝜔𝑢,𝑧(0)=𝑧₀,𝑧∈𝑍_1/2,𝑡∈(0,𝜏], (2.11) where𝑦𝑣𝑧=,𝐵_𝜔=01_𝜔,𝒜=0𝐼_𝑋−𝐴0 (2.12) is an unbounded linear operator with domain 𝐷(𝒜)=𝐷(𝐴)×𝐷(𝐴^1/2).

The proof of the following theorem follows in the same way as [4, Theorem  3.1], by putting 𝑐=0 and 𝑑=1 or directly from [5, lemma  2.1] or [6, Lemma  3.1].

Theorem 2.2. The operator 𝒜 given by (2.12) is the infinitesimal generator of a strongly continuous semigroup {𝑇(𝑡)}_𝑡∈ℝ given by 𝑇(𝑡)𝑧=^∞_𝑗=1𝑒^𝐴𝑗𝑡𝑃_𝑗𝑧,𝑧∈𝑍,𝑡≥0, (2.13) where {𝑃_𝑗}_𝑗≥1 is a complete family of orthogonal projections in the Hilbert space 𝑍_1/2 given by 𝑃_𝑗=diag𝐸_𝑗,𝐸_𝑗𝐴,𝑗≥1,_𝑗=𝑅_𝑗𝑃_𝑗,𝑅_𝑗=01−𝜆_𝑗0. (2.14)
Also, 𝐴^∗_𝑗=𝑅^∗_𝑗𝑃_𝑗,𝑅^∗_𝑗=𝜆0−1_𝑗0. (2.15) Moreover, 𝑒^𝐴𝑗𝑠=𝑒^𝑅𝑗𝑠𝑃_𝑗 and the eigenvalues of 𝑅_𝑗 are 𝜆_𝑗𝑖 and −𝜆_𝑗𝑖.

Now, before proving the main theorem, we will give the definition of approximate controllability for this system. To this end, for all 𝑧₀∈𝑍_1/2 and 𝑢∈𝐿²(0,𝜏;𝑈), the initial value problem𝑧^′=𝒜𝑧+𝐵_𝜔𝑢(𝑡)(],𝑧(0),𝑧∈𝑍,𝑡∈0,𝜏=𝑧₀, (2.16) where the control function 𝑢 belongs to 𝐿²(0,𝜏;𝑈), admits only one mild solution given by𝑧(𝑡)=𝑇(𝑡)𝑧₀+^𝑡₀𝑇(𝑡−𝑠)𝐵_𝜔𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏]. (2.17)

Definition 2.3 (Approximate Controllability). The system (2.16) is said to be approximately controllable on [0,𝜏] if for every 𝑧₀, 𝑧₁∈𝑍_1/2, 𝜀>0 there exists 𝑢∈𝐿²(0,𝜏;𝑈) such that the solution 𝑧(𝑡) of (2.17) corresponding to 𝑢 verifies 𝑧(0)=𝑧₀,𝑧(𝜏)−𝑧₁<𝜀. (2.18)

Consider the following bounded linear operator:𝐺∶𝐿²(0,𝜏;𝑍)⟶𝑍,𝐺𝑢=^𝜏₀𝑇(𝜏−𝑠)𝐵_𝜔𝑢(𝑠)𝑑𝑠, (2.19) whose adjoint operator 𝐺^∗∶𝑍→𝐿²(0,𝜏;𝑍) is given by𝐺^∗𝑧(𝑠)=𝐵^∗_𝜔𝑇^∗(𝜏−𝑠)𝑧,∀𝑠∈[0,𝜏],∀𝑧∈𝑍. (2.20) The following lemma is trivial

Lemma 2.4. The equation (2.16) is approximately controllable on [0,𝜏] if, and only if, Rang(𝐺)=𝑍.

The following result is well known from linear operator theory.

Lemma 2.5. Let 𝑊 and 𝑍 be Hilbert spaces and 𝐺^∗∈𝐿(𝑍,𝑊) the adjoint operator of the linear operator 𝐺∈𝐿(𝑊,𝑍). Then, Rang(𝐺)=𝑍⟺Ker𝐺^∗={0}. (2.21)

As a consequence of the foregoing Lemma, one can prove the following result.

Lemma 2.6. Let 𝑊 and 𝑍 be Hilbert spaces and 𝐺^∗∈𝐿(𝑍,𝑊) the adjoint operator of the linear operator 𝐺∈𝐿(𝑊,𝑍). Then, Rang(𝐺)=𝑍 if, and only if, one of the following statements holds: (a)Ker(𝐺^∗)={0},(b)⟨𝐺𝐺^∗𝑧,𝑧⟩>0, 𝑧≠0 in 𝑍,(c)_{lim+∝→0𝛼(𝛼𝐼+𝐺𝐺}^∗₎⁻¹_𝑧=0,(d)sup_𝛼>0‖𝛼(𝛼𝐼+𝐺𝐺^∗)⁻¹‖≤1.

The following theorem follows directly from (2.20) and Lemmas 2.4 and 2.6.

Theorem 2.7. The equation (2.16) is approximately controllable on [0,𝜏] iff 𝐵^∗_𝜔𝑇^∗(𝑡)𝑧=0,∀𝑡∈[0,𝜏],⟹𝑧=0. (2.22)

Now, we are ready to formulate and prove the main theorem of this work.

Theorem 2.8 (Main Theorem). If for an open nonempty set 𝜔⊂Ω the restrictions 𝜙^𝜔_𝑗,𝑘=𝜙_𝑗,𝑘|_𝜔 of 𝜙_𝑗,𝑘 to 𝜔 are linearly independent functions on 𝜔, then for all 𝜏≥2/𝜋^𝑚−1 the system (2.16) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (2.16) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗_𝜔𝑇^∗(𝜏−𝑡)𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀, (2.23) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀. (2.24)

Proof. We will apply Theorem 2.7 to prove the controllability of system (2.16). To this end, we observe that the adjoint of operator 𝐵_𝜔 is by 𝐵^∗_𝜔=01_𝜔,𝑇^∗(𝑡)𝑧=^∞_𝑗=1𝑒^{𝐴∗𝑗𝑡}𝑃_𝑗𝑧,𝑧∈𝑍,𝑡≥0. (2.25) Therefore, 𝐵^∗_𝜔𝑇^∗(𝑡)𝑧=^∞_𝑗=1𝐵^∗_𝜔𝑒^{𝑅∗𝑗𝑡}𝑃_𝑗𝑧. (2.26) On the other hand, we have that 𝑒^{𝑅∗𝑗𝑡}=cos𝜆_𝑗1𝑡−𝜆_𝑗sin𝜆_𝑗𝑡𝜆_𝑗sin𝜆_𝑗𝑡cos𝜆_𝑗𝑡=cos𝑗^𝑚𝜋^𝑚1𝑡−𝑗^𝑚𝜋^𝑚sin𝑗^𝑚𝜋^𝑚𝑡𝑗^𝑚𝜋^𝑚sin𝑗^𝑚𝜋^𝑚𝑡cos𝑗^𝑚𝜋^𝑚𝑡. (2.27) Suppose for all 𝑧∈𝑍_1/2 that 𝐵^∗_𝜔𝑇^∗(𝑡)𝑧=^∞_𝑗=1𝑗^𝑚𝜋^𝑚sin𝑗^𝑚𝜋^𝑚𝑡1_𝜔𝐸_𝑗𝑧₁+cos𝑗^𝑚𝜋^𝑚𝑡1_𝜔𝐸_𝑗𝑧₂=0,∀𝑡∈[0,𝜏]. (2.28) Then, if we make the change of variable 𝑠=𝜋^𝑚𝑡, we obtain that ^∞_𝑗=1𝑗^𝑚𝜋^𝑚sin𝑗^𝑚𝑠1_𝜔𝐸_𝑗𝑧₁+cos𝑗^𝑚𝑠1_𝜔𝐸_𝑗𝑧₂=0,∀𝑠∈0,𝜋^𝑘𝜏. (2.29) Since 𝜏≥2/𝜋^𝑚−1 we get that ^∞_𝑗=1𝑗^𝑚𝜋^𝑚sin𝑗^𝑚𝑠1_𝜔𝐸_𝑗𝑧₁+cos𝑗^𝑚𝑠1_𝜔𝐸_𝑗𝑧₂=0,∀𝑠∈[0,2𝜋]. (2.30) On the order hand, it is well known that {1,cos(𝑛𝑠),sin(𝑛𝑠)∶𝑛=1,2,3,…} is an orthogonal base of 𝐿²[0,2𝜋], which implies that {cos(𝑗^𝑚𝑠),sin(𝑗^𝑚𝑠)∶𝑗=1,2,3,…} is an orthogonal set in 𝐿²[0,2𝜋], and therefore 1_𝜔𝐸_𝑗𝑧₁(𝑥)=0,1_𝜔𝐸_𝑗𝑧₂(𝑥)=0,∀𝑥∈Ω,𝑗=1,2,3,…, (2.31) that is, 𝐸_𝑗𝑧₁(𝑥)=0,𝐸_𝑗𝑧₂(𝑥)=0,∀𝑥∈𝜔,𝑗=1,2,3,…, (2.32) that is, ^𝛾𝑗_𝑘=1𝑧₁,𝜙_𝑗,𝑘𝜙_𝑗,𝑘(𝑥)=0,^𝛾𝑗_𝑘=1𝑧₂,𝜙_𝑗,𝑘𝜙_𝑗,𝑘(𝑥)=0,∀𝑥∈𝜔,𝑗=1,2,3,…. (2.33) Since the restrictions 𝜙^𝜔_𝑗,𝑘=𝜙_𝑗,𝑘|_𝜔 of 𝜙_𝑗,𝑘 to 𝜔 are linearly independent functions on 𝜔, we get that 𝑧₁,𝜙_𝑗,𝑘=0,𝑧₂,𝜙_𝑗,𝑘=0,𝑗=1,2,3,…;𝑘=1,2,…,𝛾_𝑗. (2.34) Therefore, 𝑃_𝑗(𝑧)=𝐸_𝑗𝑧₁𝐸_𝑗𝑧₂=00. (2.35) Hence, 𝑧=∑^∞_𝑗=1𝑃_𝑗𝑧=0, and the proof of the approximate controllability of the system (2.16) is completed.
Now, given the initial and the final states 𝑧₀ and 𝑧₁, we consider the sequence of controls 𝑢_𝛼(⋅)=𝐵^∗_𝜔𝑇^∗(𝜏−⋅)𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀=𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀,𝛼>0. (2.36) Then, 𝐺𝑢_𝛼=𝐺𝐺^∗𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀=𝛼𝐼+𝐺𝐺^∗−𝛼𝐼𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀=𝑧₁−𝑇(𝜏)𝑧₀−𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀. (2.37) From part (c) of Lemma 2.6, we know that lim_𝛼→0+𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀=0. (2.38) Therefore, lim_𝛼→0+𝐺𝑢_𝛼=𝑧₁−𝑇(𝜏)𝑧₀, (2.39) that is, lim_𝛼→0+𝑇(𝜏)𝑧₀+^𝜏₀𝑇(𝜏−𝑠)𝐵_𝜔𝑢_𝛼(𝑠)𝑑𝑠=𝑧₁. (2.40) This completes the proof of the theorem.

The following basic theorem will be used to prove an important consequence of the foregoing theorem.

Theorem 2.9 (see [7,  Theorem  1.23,  page  20]). Suppose Ω⊂ℝ^𝑛 is open, nonempty, and connected set, and 𝑓 is real analytic function in Ω with 𝑓=0 on a nonempty open subset 𝜔 of Ω. Then, 𝑓=0 in Ω.

Corollary 2.10. If 𝜙_𝑗,𝑘 are analytic functions on Ω, then for all open nonempty set 𝜔⊂Ω and all 𝜏≥2/𝜋^𝑚−1 the system (2.16) is approximately controllable on [0,𝜏].

Proof. It is enough to prove that, for all open nonempty set 𝜔⊂Ω the restrictions 𝜙^𝜔_𝑗,𝑘=𝜙_𝑗,𝑘|_𝜔 of 𝜙_𝑗,𝑘 to 𝜔 are linearly independent functions on 𝜔, which follows directly from Theorem 2.9.

3. Applications

For the applications, we will use Corollary 2.10 and the following fact.

Theorem 3.1 (see [3]). The eigenfunctions of the operator −Δ with Dirichlet boundary conditions on Ω are real analytic functions in Ω.

In this section, we will prove the approximate controllability of (1.7) and (1.8). Specifically, we will prove the following theorem.

Theorem 3.2. For all open nonempty set 𝜔⊂Ω=[0,1], we have the following statements. (a)For all 𝜏≥2 the system (1.7) is approximately controllable on [0,𝜏].(b)For all 𝜏≥2/𝜋 the system (1.8) is approximately controllable on [0,𝜏].

Proof. Let 𝑋=𝐿²(Ω) and consider the linear unbounded operator −Δ∶𝐷(−Δ)⊂𝑋→𝑋 with 𝐷(−Δ)=𝐻¹₀(Ω)∩𝐻²(Ω). In this case, the eigenvalues and the eigenfunctions of 𝐴=−Δ are given, respectively, by 𝜆_𝑗=𝑗²𝜋²,𝜙_𝑗(𝑥)=√2sin𝑗𝜋𝑥,𝑗=1,2,3,…,𝐴𝑥=−Δ𝑥=^∞_𝑗=1𝜆_𝑗𝑥,𝜙_𝑗𝜙_𝑗=^∞_𝑗=1𝑗²𝜋²𝑥,𝜙_𝑗𝜙_𝑗. (3.1) Then, 𝑚=1 and 𝜏≥2. So, (a) follows from Corollary 2.10.
To prove (b), we consider the operator 𝐴𝑥=(−Δ)²𝑥=^∞_𝑗=1𝜆²_𝑗𝑥,𝜙_𝑗𝜙_𝑗=^∞_𝑗=1𝑗⁴𝜋⁴𝑥,𝜙_𝑗𝜙_𝑗. (3.2) Then, 𝑚=2 and 𝜏≥2/𝜋. So, (b) follows from Corollary 2.10.

4. Final Remark

The result presented in this paper can be formulated in a more general setting. Indeed, we can consider the following second-order evolution equation in a general Hilbert space 𝑋:̈(𝑡)(],𝑦(0)𝑦+𝐴𝑦=𝐶𝑢,𝑡∈0,𝜏=𝑦₀,𝑦̇(0)=𝑦₁, (4.1) where, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is an unbounded linear operator in 𝑋 with the spectral decomposition given by (1.2), the control 𝑢∈𝐿²(0,𝜏;𝑋) and 𝐶∶𝑋→𝑋 is a linear and bounded operator (linear and continuous). In this case, the characteristic function set is a particular operator 𝐶, and the following theorem is a generalization of Theorem 2.8.

Theorem 4.1. If the vectors 𝐶^∗𝜙_𝑗,𝑘 are linearly independent in 𝑋, then for all 𝜏≥2/𝜋^𝑚−1 the system (4.1) is approximately controllable on [0,𝜏]. Moreover, a sequence of controls steering the system (2.16) from initial state 𝑧₀ to an 𝜖 neighborhood of the final state 𝑧₁ at time 𝜏>0 is given by 𝑢_𝛼(𝑡)=𝐵^∗𝑇^∗(𝜏−𝑡)𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀, (4.2) and the error of this approximation 𝐸_𝛼 is given by 𝐸_𝛼=𝛼𝛼𝐼+𝐺𝐺^∗−1𝑧₁−𝑇(𝜏)𝑧₀, (4.3) where the operator 𝐵 is given by 𝐵=[⁰_𝐶].

The novelty of this result is based on the fact that, it is general, rigorous, applicable, and easily comprehensible by those young mathematician who are located in places away from majors research center.

Acknowledgment

This work was supported by the CDCHT-ULA-project: C-1667-09-05-AA and BCV.